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Rh IIISTOPvY.] to have placed their aathors at the head of living mathe maticians. The reader will find the subject admirably treated in Baltzer s Theorie und Amvendung der Deter- minenten; and more briefly in Salmon s Higher Algebra. Elementary treatises have also been published by Spottis- woode in 1851, by Brioschi in 1854, by Todhunter in his Theory of Equations in 1861, and by Dodgson in 1867. iiilian The attention of the learned has, during the present igebni. century, been called to a branch of the history of algebra, in no small degree interesting ; we mean the cultivation of the science to a considerable extent, and at a remote period, in India. We are indebted, we believe, to Mr Reuben Burrow for some of the earliest notices which reached Europe on this very curious subject. His eagerness to illustrate the history of the mathematical sciences led him to collect oriental manuscripts, some of which, in the Persian language, with partial translations, were bequeathed to his friend Mr Dalby of the Royal Military College, who communicated them to such as took an interest in the subject, about the year 1800. In the year 1813, Mr Edward Strachey published in this country a translation from the Persian of the Bija Ganita (or Vija Ganita}, a Hindoo treatise on algebra; and in 1816 Dr John Taylor published at Bombay a translation of Lelawati (or Lilavati), from the Sanscrit original. This last is a treatise on arithmetic and geometry, and both are the production of an oriental algebraist, Bhascara Acharya. Lastly, in 1817, there came out a work entitled Algebra, Arithmetic, and Mensuration, from the Sanscrit of Brahme gupta and Bhascara, translated by Henry Thomas Cole brooke, Esq. This contains four different treatises, origi nally written in Sanscrit verse, viz., the Vija Ganita and Lilavati of Bhascara Acharya, and the Ganitad haya and Cuttacad hyaya of Brahmegupta. The first two form the preliminary portion of Bhascara s Course of Astronomy, entitled tiidd hanta Siromani, and the last two are the twelfth and eighteenth chapters of a similar course of astronomy, entitled Brahma-siddhanta. The time when Bhascara wrote is fixed with great pre cision, by his own testimony and other circumstances, to a date that answers to about the year 1150 of the Chris tian era. The works of Brahmegupta are extremely rare, and the age in which he lived is less certain. Mr Davis, an oriental scholar, who first gave the public a correct view of the astronomical computations of the Hindoos, is of opinion that he lived in the 7th century ; and Dr William Hunter, another diligent inquirer into Indian science, assigns the year 628 of the Christian era as about the time he flourished. From various arguments, Mr Colebrooke concludes that the age of Brahmegupta was antecedent to the earliest dawn of the culture of the sciences among the Arabians, so that the Hindoos must have possessed algebra before it was known to that . nation. Brahmegupta s treatise is not, however, the earliest work known to have been written on this subject. Ganessa, a distinguished astronomer and mathematician, and the most eminent scholiast of Bhascara, quotes a passage from a much older writer, Arya-Bhatta, specifying algebra under the designation of Vija, and making separate mention of Cuttaca, a problem subservient to the resolution of inde terminate problems of the first degree. He is understood by another of Bhascara s commentators to be at the head of the older writers. They appear to have been able to resolve quadratic equations by the process of completing the square; and hence Mr Colebrooke presumes that the treatise of Arya-Bhatta then extant extended to quadratic equations in the determinate analvsis. and to indeterminate 517 equations of the first degree, and probably to those of the second. The exact period when Arya-Bhatta lived cannot bo determined with certainty; but Mr Colebrooke thinks it probable that this earliest of known Hindoo algebraists wrote as far back as the fifth century of the Christian era, and perhaps earlier. He lived therefore nearly as early as the Grecian algebraist Diophantus, who is reckoned to have flourished in the time of the emperor Julian, or about A.D. 360. Mr Colebrooke has instituted a comparison between the Indian algebraist and Diophantus, and found reason to conclude that in the whole science the latter is very far behind the former. He says the points in which the Hindoo algebra appears particularly distinguished from the Greek are, besides a better and more convenient algo rithm, 1st, the management of equations of more than one unknown quantity; 2d, the resolution of equations of a higher order, in which, if they achieved little, they had at least the merit of the attempt, and anticipated a modern discovery in the resolution of biquadratics; 3d, general methods for the resolution of indeterminate pro blems of the first and second degrees, in which they went far indeed beyond Diophantus, and anticipated discoveries of modern algebraists ; and 4th, the application of algebra to astronomical investigations and geometrical demonstration, in which also they hit upon some matters which have been re-invented in modern times. When we consider that algebra made little or no pro gress among the Arabians, a most ingenious people, and particularly devoted to the study of the sciences, and that centuries elapsed from its first introduction into Europe until it reached any considerable degree of perfection, we may reasonably conjecture that it may have existed in one shape or other in India long before the time of Arya- Bhatta; indeed, from its close connection with their doc trines of astronomy, it may be supposed to have descended from a very remote period along with that science. Pro fessor Playfair, adopting the opinion of Bailly, the eloquent author of the Astronomic Indienne, with great ingenuity attempted to prove, in a Memoir on the Astronomy of the Brahmins, that the observations on which the Indian astronomy is founded were of great antiqxiity, indeed more than 3000 years before the Christian era. The very remote origin of the Indian astronomy had been strongly ques tioned by many in this country, and also on the Continent; particularly by Laplace, and by Delambre in his Ilistoire de V Astronomic Ancienne, tome i. p. 400, &c., and again in Ilistoire de V Astronomic du Moyen Age, Discours Prelimi* naire, p. 18, &c., where he speaks slightingly of their algebra; and in this country, Professor Leslie, in his Philosophy of Arithmetic, pp. 225 and 226, calls the Lilavati &quot; a very poor performance, containing merely a few scanty precepts couched in obscure memorial verses.&quot; We are disposed to agree with Professor Leslie as to the value, and with Professor Playfair as to the antiquity of this Hindoo algebra. That it should have remained in a state of infancy for so many centuries is accounted for by the latter author in the following passage : &quot; In India everything [as well as algebra] seems equally insurmountable, and truth and error are equally assured of permanence in the stations they have once occupied. The politics, the laws, the religion, the science, and the manners, seem all nearly the same as at the remotest period to which history extends. Is it because the power which brought about a certain degree of civilisation, and advanced science to a certain height, has either ceased to act, or has met with such a resistance as it is barely able to overcome ] or is it because the dis- coveries which the Hindoos arc in possession of are an inheritance from some more inventive and more ancient